Decoding apparatus, dequantizing method, distribution determining method, and program thereof

ABSTRACT

A decoding apparatus includes a random-number generating section and a decoding section. The random-number generating section generates random numbers according to distribution of original data corresponding to respective quantization indexes. The decoding section generates decoded data on a basis of the random numbers generated by the random-number generating section.

This is a Division of application Ser. No. 11/179,988 filed Jul. 13,2005. The disclosure of the prior application is hereby incorporated byreference herein in its entirety.

BACKGROUND OF THE INVENTION Field of the Invention

The invention relates to a decoding apparatus for decoding code datagenerated by an encoding process. More specifically, the inventionrelates to a decoding apparatus for dequantize code data generated by anencoding process including quantizing data, to decode the code data.

Since images, audios or the like have enormous amount of data, it iscommon to reduce an amount of the data by compressing it and then storeor transmit the compressed data. For example, an amount of multi-valueimage data generated when color documents or photographs are transformedinto an electronic form by a scanner or when scenery is photographed bya digital camera can be significantly reduced by compressing the datausing a lossy coding process such as JPEG, JPEG200 or the like.

However, the lossy coding process causes coding distortion; this is aproblem. In particular, the JPEG process has a problem in that blockdistortion occurs at DCT block boundaries of decoded images (codingdistortion).

In this connection, a generation mechanism for the coding distortion ofthe lossy coding process will be first described.

FIGS. 1A and 1B are block diagrams schematically illustrating atransform coding method such as JPEG and JPEG2000, where FIG. 1A showsan outline of an encoding process and FIG. 1B shows an outline of adecoding process.

FIGS. 2A to 2C are diagrams illustrating a quantization process in thetransform coding method. A transform coefficient T(c, i, j) and aquantization index Q(c, i, j) shown in FIGS. 1A and 1B are functions ofvariables c, i and j. The variable c is an index indicating a kind oftransform coefficient. For example, in the case of the DCT transformusing 8×8 blocks, the variable c is a value (an integer in a range of 1to 64) indicating one of 64 (8×8) transform coefficients. In a case ofthe wavelet transform, the variable c is a value indicating one ofcomponents such as 1HH, 1LH, 1HL, 2HH, 2LH, 2HL, . . . , NLL. Inaddition, the transform variables i and j are variables indicatingpositions of the transform coefficients, respectively. For example, inthe case of the DCT transform, a c-th transform coefficient in a blocklocated at an i-th row from the top and a j-th column from the left isindicated as T(c, i, j). In the case of the wavelet transform, data of ac-th transform coefficient located at an i-th row from the top and aj-th column from the left is indicated as T (c, i, j).

As shown in FIG. 1A, in an encoding process of the transform codingmethod, an input image G is subject to a transform process such as thediscrete cosine transform (DCT) or the wavelet transform to generate atransform coefficient T of the input image G. The transform coefficientT is then quantized into a quantization index Q. The quantization indexQ is subject to an entropy coding process (lossless coding process) tobe a compression code F.

Here, the quantization index refers to information used to distinguishquantization values. In addition, the quantization value refers to avalue to which a group of numerical values within a specific range(quantization interval) are degenerated. For example, as shown in FIGS.2A to 2C, the quantization values are discrete values (−2×D(c) to 2×D(c)in this example) representing quantization intervals (A-2˜A2),respectively.

Code data (the compression code F) generated in this way areentropy-decoded into a quantization index Q, as shown in FIG. 1B. Thisquantization index Q is equivalent to the quantization index Q in theencoding process.

Then, the quantization index Q is dequantized into a transformcoefficient R (i.e., a dequantization value). Thereafter, the transformcoefficient R is inversely transformed to generate a decoded image H.

Here, the dequantization value refers to a value, which is generatedbased on the quantization index or the quantization value and is usedfor decoding of data. For example, the dequantization value is atransform coefficient of the JPEG or JPEG2000 (transform coefficientbeing associated with a quantization index).

In the above-described process, coding distortion occurs during thequantization. In general, precision of the transform coefficient T of anoriginal image is higher than that of the quantization index Q.Accordingly, the transform coefficient R reproduced by using thequantization index Q may be different from the original transformcoefficient T. This is the cause of the coding distortion.

Next, the quantization and the dequantization will be described indetail with reference to FIGS. 2A to 2C.

The quantization is performed using a quantization step width D(c)prepared for each transform coefficient c. The quantization step widthD(c) is a function of the kind of transform coefficient c. For example,in the case of JPEG, the quantization index Q is calculated according tothe following equation in the quantization.

Q(c,i,j)=round(T(c,i,j)/D(c))

Where “round( )” is a function outputting an integer closest to an inputvalue.

In addition, the dequantization value R is calculated according to thefollowing equation in the dequantization.

R(c,i,j)=Q(c,i,j)×D(c)

In the case of JPEG2000, the quantization index Q and the dequantizationvalue R are calculated according to the following equations.

Q(c,i,j)=sign(T(c,i,j))×floor(|T(c,i,j)|/D(c))

R(c,i,j)=(Q(c,i,j)+r)×D(c), if Q(c, i, j)>0

R(c,i,j)=(Q(c,i,j)−r)×D(c), if Q(c, i, j)<0

R(c,i,j)=0, if Q(c, i, j)=0

Where, “sign ( )” is a function outputting positive sign or negativesign, “floor ( )” is a function nulling decimal places, and “∥” is asymbol representing an absolute value.

In addition, “r” is a numerical value in a range of 0 to 1, typicallyr=0.5. In the JPEG2000, there may be a case where lower bits are notencoded. Here, a case where all bits including the least significant bitare encoded will be described by way of examples. Alternatively, in theJPEG2000, it is possible to obtain number of bits, which are not encodedin the encoding, from a code stream in the decoding. Accordingly, byshifting the quantization step width D to the left by the number of bitsand setting the shifted quantization step width as a new quantizationwidth, the JPEG2000 may have the same operation as the JPEG.

As shown in FIG. 2A, in the encoding process of the JPEG, transformcoefficients T (before the quantization) generated by the transformprocess performed for the input image G are distributed on an X axis,which is a numerical straight line.

If a transform coefficient T exists in a quantization interval A0, thequantization index Q becomes 0 by the quantization process. Similarly,if a transform coefficient T exists in a quantization interval Aq, thequantization index Q becomes q.

Then, when the dequantization is performed for the quantization index Q,in a case in which the quantization index Q is 0, the dequantizationvalue R of 0 is generated by the dequantization process. In a case inwhich the quantization index Q is 1, the dequantization value R of D(c)is generated.

Similarly, in the JPEG2000, as shown in FIG. 2B, if a transformcoefficient T exists in a quantization interval Aq, the quantizationindex Q becomes q. Then, when the dequantization is performed for thequantization index Q, dequantization values corresponding toquantization indexes Q in a one-to-one manner are generated.

Here, for the sake of simplicity, only the quantization interval Aq inwhich the quantization index Q becomes q will be considered.

It is assumed that the transform coefficient T exists in thequantization interval Aq.

As shown in FIG. 2C, the quantization interval Aq has a range of d1 tod2. In this case, the transform coefficient T is included in the rangeof d1 to d2. In addition, it is assumed that a dequantization value ofthe transform coefficient T is R.

Under this condition, a transform coefficient for generating a decodedimage is the dequantization value R. However, the transform coefficientT of an original image has any value within the range of d1 to d2 and isnot always equivalent to the dequantization value R. At this time, adifference between the original transform coefficient T and thedequantization value R occurs. This difference is the cause of thecoding distortion.

As described previously, the lossy coding process realizes a lossy datacompression by degenerating a plurality of data values (original datavalues existing in each quantization interval) into one quantizationvalue (a quantization value corresponding to each quantizationinterval), but at the same time, the coding distortion occurs due to thequantization.

In order to reduce this coding distortion, a parameter for reducingcompression efficiency in the encoding process may be selected.

However, this causes a problem that encoding efficiency is reduced andthe amount of data is increased.

Further, when previously encoded data is intended to be high-qualityimages, it is impossible to employ such a process in which thecompression efficiency is reduced.

For this reason, various techniques have been suggested for overcomingthe image distortion problem in the decoding process.

In a broad classification, there are two types of method, that is, afiltering method and a noise method. In the filtering method, a decodedimage is subject to a lowpass filtering process so as to make codingdistortion faint and be conspicuous. In the noise method, noises areadded to the decoded image or the transformation coefficient so as tomake coding distortion faint and be conspicuous.

First, the method using the low pass filtering process (the filteringmethod) will be described.

For example, it is known to provide a method applying a low pass filterto only a boundary between DCT blocks in order to remove blockdistortion.

This method makes the coding distortion faint using the low pass filterso that it is difficult for this distortion to be discriminated.

However, this method has a problem in that edge components of anoriginal image become faint as well.

In addition, it is known to provide a method, which prepares a pluralityof low pass filters, determines as to whether or not edges are presentin an image, and selectively applies a filter not to cause the edges tobe faint, based on a determination result.

Next, the method of adding noises (the noise method) will be described.

For example, it is known to provide a method, which adds noises to DCTcoefficients so as to make the coding distortion faint, when it isdetermined that distortion is noticeable in a region.

In this method, the coding distortion is considered to be noticeablewhen the region is determined to be a flat image region.

When a decoded image is generated from an encoded image (i.e., adecoding process is performed), it is a goal to approach the decodedimage as close as possible to an original image before the originalimage is subject to an encoding process.

From this point of view, the methods according to prior art do notprovide an optimal solution since faintness of the image by the low passfilter or addition of the noises does not approach the decoded image tothe original image.

More specifically, these methods may have some side effects as follows.

(1) In the filtering method, signals in a high-frequency band of thedecoded image are suppressed. Accordingly, when textures of highfrequency components are present in the original image, it is impossibleto reproduce these textures.

(2) In the filtering method, there may be a possibility of dullness ofthe edge due to a possibility of incorrect edge determination.

(3) In the noise method, there may be a possibility of texturesoccurring, which are not present in the original image, due to theaddition of noises.

SUMMARY OF THE INVENTION

According to one aspect of the invention, a decoding apparatus includesa random-number generating section and a decoding section. Therandom-number generating section generates random numbers according todistribution of original data corresponding to respective quantizationindexes. The decoding section generates decoded data on a basis of therandom numbers generated by the random-number generating section.

According to another aspect of the invention, a decoding apparatusincludes a standard-deviation acquiring section, a multiplying section,an upper-limit-value acquiring section, and a random-number generatingsection. The standard-deviation acquiring section acquires a standarddeviation of transform coefficients corresponding to quantizationindexes. The multiplying section multiplies the standard deviationacquired by the standard-deviation acquiring section by a preset value.The upper-limit-value acquiring section acquires an upper limit value ofgenerated random numbers. The random-number generating section uniformlygenerates the random numbers, with smaller one of the standard deviationmultiplied by the preset value by the multiplying section and the upperlimit value acquired by the upper-limit-value acquiring section being anupper limit.

According to another aspect of the invention, a decoding apparatusincludes a frequency measuring section, a histogram normalizing section,an addition range determining section, and a distribution determiningsection. The frequency measuring section measures appearance frequencyof quantization indexes. The histogram normalizing section generatesnormalized histograms based on the appearance frequency measured by thefrequency measuring section. The addition range determining sectiondetermines an addition range in which frequency distribution of thequantization indexes is added. The distribution determining sectiondetermines at least one of a standard deviation and a variance based onthe histograms generated by the histogram normalizing section and theaddition range determined by the addition range determining section.

According to another aspect of the invention, a decoding apparatusincludes a first calculating section, a second calculating section, anda distribution estimating section. The first calculating sectioncalculates at least one of a standard deviation of quantization indexesand a variance of the quantization indexes. The second calculatingsection calculates at least one of a standard deviation of a Laplacedistribution and a variance of the Laplace distribution so that a sum ofvalues of frequency of the quantization indexes within a preset range isequal to an integral value of a Laplace distribution functioncorresponding to the preset range. The distribution estimating sectionestimates distribution of original data corresponding to thequantization indexes using at least one of (A) the at least one of thestandard deviation and the variance calculated by the first calculatingsection and (B) the at least one of the standard deviation and thevariance calculated by the second calculating section.

According to another aspect of the invention, a dequantizing methodincludes generating random numbers according to distribution of originaldata corresponding to respective quantization indexes; and generatingdequantization values based on the generated random numbers.

According to another aspect of the invention, a distribution determiningmethod includes adding values of frequency of the respectivequantization indexes; and calculating at least one of a variance of aLaplace distribution and a standard deviation of the Laplacedistribution so that a resultant value of the addition of values offrequency is equal to an integral value when a Laplace distributionfunction is integrated so that an integral range of a right side isequal to an integral range of a left side with using a maximum frequencyposition of the Laplace distribution as a reference.

According to another aspect of the invention, a storage medium, which isreadable by a computer, stores a program of instructions executable bythe computer to perform a dequantization function comprising the stepsof generating random numbers according to distribution of original datacorresponding to respective quantization indexes; and generatingdequantization values based on the generated random numbers.

According to another aspect of the invention, a storage medium, which isreadable by a computer, stores a program of instructions executable bythe computer to perform a dequantization function comprising the stepsof adding values of frequency of the respective quantization indexes;and calculating at least one of a variance of a Laplace distribution anda standard deviation of the Laplace distribution so that a resultantvalue of the addition of values of frequency is equal to an integralvalue when a Laplace distribution function is integrated so that anintegral range of a right side is equal to an integral range of a leftside with using a maximum frequency position of the Laplace distributionas a reference.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiment of the invention will be described in detail based on thefollowing figures, wherein:

FIG. 1A is a block diagram schematically illustrating an encodingprocess of a transform coding method such as JPEG and JPEG2000;

FIG. 1B is a block diagram schematically illustrating a decoding processof a transform coding method such as JPEG and JPEG2000;

FIG. 2A is a diagram illustrating a quantization process in thetransform coding method;

FIG. 2B is a diagram illustrating a quantization process in thetransform coding method;

FIG. 2C is a diagram illustrating a quantization process in thetransform coding method;

FIG. 3 is a diagram illustrating a hardware configuration of a decodingapparatus 2 to which a decoding method according to embodiments of theinvention is applied, with providing a controller 20 centrally;

FIG. 4 is a diagram illustrating a functional configuration of adecoding program 5 executed by the controller 20 shown in FIG. 3, forimplementing a decoding method according to the embodiments of theinvention;

FIG. 5 is a diagram illustrating a distribution estimating section 520(FIG. 4) in more detail;

FIG. 6 is a diagram illustrating an exemplary histogram h and anexemplary distribution function L (Laplace function);

FIG. 7A is a diagram illustrating a distribution estimating processperformed by a zero transform coefficient distribution estimatingsection 526;

FIG. 7B is a diagram illustrating a distribution estimating processperformed by a zero transform coefficient distribution estimatingsection 526;

FIG. 8A is a schematic diagram illustrating correction by a correctingsection 580;

FIG. 8B is a schematic diagram illustrating correction by a correctingsection 580;

FIG. 9 is a flow chart of a decoding process S10 by the decoding program5 (FIG. 4);

FIG. 10 is a diagram illustrating an exemplary filter used by adequantization-value estimating section 500.

FIG. 11A is a diagram illustrating a polygonal-line approximation;

FIG. 11B is a diagram illustrating a polygonal-line approximation;

FIG. 11C is a diagram illustrating a polygonal-line approximation;

FIG. 11D is a diagram illustrating a polygonal-line approximation;

FIG. 11E is a diagram illustrating a polygonal-line approximation;

FIG. 12 is a diagram illustrating transform coefficients in theJPEG2000;

FIG. 13 is a diagram illustrating a distribution estimating section 520in a second embodiment in more detail;

FIG. 14 is a graph showing a relationship between the maximum value ofquantization values and an optimal N value;

FIG. 15 is a flow chart of an N value determining process;

FIG. 16 is a graph showing a ratio of a standard deviation of originaltransform coefficients to an estimated standard deviation; and

FIG. 17 is a diagram showing a ratio of a standard deviation of originaltransform coefficients to an estimated standard deviation when a fourthmodification is applied.

DETAILED DESCRIPTION OF THE EMBODIMENTS OF THE INVENTION

Hereinafter, a first embodiment of the invention will be described.

In this embodiment, a case where code data encoded according to the JPEGis decoded will be described by way of examples. A decoding process tobe described in this embodiment is approximately similar to thatdescribed in ITU-T Recommendation T.81. However, the decoding process ofthis embodiment is different in a dequantization process from that ofITU-T Recommendation T.81.

[Hardware Configuration]

First, a hardware configuration of the decoding apparatus 2 according tothis embodiment will be described.

FIG. 3 is a diagram illustrating a hardware configuration of thedecoding apparatus 2 to which a decoding method according to theinvention is applied, with a controller 20 as the central figure.

As shown in FIG. 3, the decoding apparatus 2 includes a controller 20including CPU 202, a memory 204 and the like, a communication unit 22, astorage unit 24 such as HDD, CD and the like, and a user interface unit(UI unit) 26 including an LCD display device or a CRT display device, akey board, a touch panel and the like.

The decoding apparatus 2 is a general-purpose computer in which adecoding program 5, which will be described later, is installed. Thedecoding apparatus 2 acquires code data through the communication unit22, the storage unit 24 or the like and decodes the acquired code data.

[Decoding Program]

FIG. 4 is a diagram illustrating a functional configuration of thedecoding program 5 executed by the controller 20 shown in FIG. 3, forimplementing a decoding method according to embodiments of theinvention.

As shown in FIG. 4, the decoding program 5 includes an entropy decodingsection 40, a dequantizing section 50 and an inverse transformingsection 60.

Also, the dequantizing section 50 includes a dequantization-valueestimating section 500, a distribution estimating section 520, anexpected-value estimating section 540, a random-number generatingsection 560, a correcting section 580, and a dequantization-valueoutputting section 590.

In the decoding program 5, the entropy decoding section 40entropy-decodes input code data and outputs the decoded data to thedequantizing section 50.

The entropy decoding section 40 of this embodiment decodes the inputcode data to generate a quantization index Q and outputs the generatedquantization index Q to the dequantizing section 50.

The dequantizing section 50 generates a dequantization value R based onthe quantization index Q input from the entropy decoding section 40 andoutputs the generated dequantization value R to the inverse transformingsection 60.

The inverse transforming section 60 performs an inverse transform basedon the dequantization value R input from the dequantizing section 50 togenerate a decoded image.

In the dequantizing section 50, the dequantization-value estimatingsection 500 estimates a dequantization value based on the quantizationindex Q input from the entropy decoding section 40, and outputs theestimated dequantization value to the correcting section 580. That is,the dequantization-value estimating section 500 does not always generatea single dequantization value for one quantization index value, but cangenerate a plurality of different dequantization values for onequantization index value. In other words, although thedequantization-value estimating section 500 generates one dequantizationvalue for each quantization index, the dequantization-value estimatingsection 500 does not necessarily generate the same dequantization valueeven when input quantization indexes have the same value.

The dequantization-value estimating section 500 of this embodimentcalculates a correction factor α of the dequantization value Rcorresponding to the quantization index of a given block, based on thequantization index of the given block and the quantization index(limited to one having a quantization index of the same kind c as thetransform coefficient) of another block adjacent to the given block, andoutputs the calculated correction factor α to the correcting unit 580.

Further, in the following description, a correction factor αcorresponding to each transform coefficient kind c and each quantizationindex q is denoted by αycq. In addition, assuming that the number ofsignals each having the transform coefficient kind c and thequantization index q is K, and that each correction factor is denoted byαycq(k) (where, k=1, 2, . . . , K).

The distribution estimating section 520 estimates distribution oftransform coefficients (of original data) based on a plurality ofquantization indexes (or, dequantization values corresponding to theplurality of quantization indexes) input from the entropy decodingsection 40, and then outputs distribution data representing theestimated distribution of transform coefficients to the expected-valueestimating section 540 and the random-number generating section 560.

The distribution estimating section 520 in this example calculates thefrequency distribution of quantization indexes for each transformcoefficient kind c, and then generates the distribution data for eachtransform coefficient kind c based on the calculated frequencydistribution.

The expected-value estimating section 540 calculates expected values ofthe dequantization values based on the distribution data input from thedistribution estimating section 520, and then outputs the calculatedexpected values and the distribution data to the correcting section 580.

More specifically, the expected-value estimating section 540 calculatesexpected values for each quantization interval (that is, expected valuesfor each quantization index value) based on the distribution datagenerated for each transform coefficient kind c.

When the transform coefficient kind is c and the quantization index Q(c,i, j) is equal to q, an expected value is indicated by E(αTcq). That is,the expected value E(αTcq) indicates estimated expected values ofdifferences between the dequantization values R corresponding to thequantization indexes in a one-to-one manner and the original transformcoefficients T corresponding to the quantization indexes.

The random-number generating section 560 generates random numbersaccording to the distribution data input from the distributionestimating section 520, and outputs the generated random numbers to thedequantization-value outputting section 590.

The correcting section 580 corrects the dequantization value (thecorrection factor α of the dequantization value in this example) inputfrom the dequantization-value estimating section 500 based on thedistribution data or the expected values input from the expected-valueestimating section 540.

Further, the correcting section 580 corrects the dequantization value(the correction factor α of the dequantization value in this example)input from the dequantization-value estimating section 500 to be in apreset range (for example, in the case of the dequantization value, aquantization interval corresponding to the quantization index), and thenoutputs the corrected dequantization value (the correction factor α) tothe dequantization-value outputting section 590.

The correcting section 580 in this example corrects the correctionfactor α input from the dequantization-value estimating section 500based on the expected value input from the expected-value estimatingsection 540 such that the frequency distribution of quantization indexescalculated by the distribution estimating section 520 becomesapproximately identical with the frequency distribution ofdequantization values calculated by the dequantization-value estimatingsection 500 for each transform coefficient kind c and each quantizationinterval, and then linearly corrects the corrected correction factor αagain to fall within a range of −0.5 to 0.5 in the JPEG.

The linear correction executed by the correcting section 580 is, forexample, achieved by selecting the maximum value αmax and the minimumvalue αmin from among the correction factors α corresponding to the samequantization index and then by linearly transforming all the correctionfactors α such that the selected maximum value αmax and minimum valueαmin fall within the preset range (the range of −0.5 to 0.5 in theJPEG).

Furthermore, the correcting section 580 may take the correction factorsα as a boundary value of this range (i.e., one of −0.5 and 0.5, which iscloser to α) if the correction factors α is outside the range of −0.5 to0.5. Also, the correcting section 580 may take the correction factors αas 0 if the correction factors α is outside the range of −0.5 to 0.5.

In addition, the JPEG2000 is different from the JPEG only in the rangeof the correction factors α. That is, in the JPEG2000, the correctingsection 580 corrects the correction factors α on the basis of a range of0≦r+α≦1 if Q(c, i, j)>0, a range of −1≦r+α≦0 if Q(c, i, j)<0, and arange of −1≦α≦1 if Q(c, i, j)=0, respectively.

The dequantization-value outputting section 590 determines adequantization value to be applied by using the dequantization value(the correction factors α of the dequantization value in this example)input from the correcting section 580 or the random numbers input fromthe random-number generating section 560, and then outputs thedetermined dequantization value to the inverse transforming section 60.

The dequantization-value outputting section 590 in this examplecalculates the dequantization value based on the correction factors αinput from the correcting section 580 or the random-number generatingsection 560 and the quantization index (or the dequantization valueassociated with the quantization index). More specifically, thedequantization-value outputting section 590 calculates thedequantization value Ry(c, i, j) to be applied, using the followingequation.

Ry(c,i,j)={Q(c,i,j)+α(c,i,j)}×D(c)

That is, the decoding program 5 of this embodiment does not apply therandom numbers generated by the random-number generating section 560 asthe dequantization values themselves, but applies the random numbersgenerated by the random-number generating section 560 as the correctionfactors α of the dequantization values.

[Distribution Estimating Section]

FIG. 5 is a diagram illustrating a distribution estimating section 520of FIG. 4 in more detail.

As shown in FIG. 5, the distribution estimating section 520 includes azero determining section 522, a non-zero transform coefficientdistribution estimating section 524, and a zero transform coefficientdistribution estimating section 526.

In the distribution estimating section 520, the zero determining section522 classifies the quantization indexes input from the entropy decodingsection 40 according to an attribute (for example, the kind of transformcoefficients) of original data corresponding to the quantizationindexes, and determines as to whether or not the frequency distributionof the original data can be estimated only by groups of quantizationindexes classified according to attributes of the original data (inother words, whether or not the frequency distribution can be estimatedusing correlation between a group of quantization indexes classifiedaccording to an attribute of the original data and another group ofquantization indexes classified according to a different attribute ofthe original data).

The zero determining section 522 in this example determines as towhether the quantization indexes input from the entropy decoding section40 correspond to zero transform coefficients or non-zero transformcoefficients, outputs the quantization indexes determined to correspondto the non-zero transform coefficients to the non-zero transformcoefficient distribution estimating section 524, and instructs the zerotransform coefficient distribution estimating section 526 to apply adistribution estimation process to the quantization indexes determinedto correspond to the zero transform coefficients, using distribution ofthe non-zero transform coefficients.

Here, the non-zero transform coefficients refer to transformcoefficients in which any of the quantization indexes of one transformcoefficient kind cis not zero. In addition, the zero transformcoefficients refer to transform coefficients in which all of thequantization indexes of one transform coefficient kind c are zero. Inother words, all transform coefficients other than the zero transformcoefficients are the non-zero transform coefficients.

The non-zero transform coefficient distribution estimating section 524estimates the frequency distribution (transform coefficients in thisexample) of the original data based on the quantization indexes inputfrom the zero determining section 522.

More specifically, the non-zero transform coefficient distributionestimating section 524 generates the frequency distribution of thegroups of quantization indexes having the same attribute (in thisexample, a plurality of quantization indexes corresponding to the sametransform coefficient kind c), and prepares a probability densityfunction of quantization indexes based on the generated frequencydistribution of quantization indexes. This probability density functionis applied as an approximation to a probability density function oftransform coefficients.

The non-zero transform coefficient distribution estimating section 524in this example prepares a histogram hc(q) of the quantization indexesQ(c, i, j) input from the zero determining section 522 (the quantizationindexes corresponding to the non-zero transform coefficients) for eachtransform coefficient kind c.

For example, the non-zero transform coefficient distribution estimatingsection 524 defines a function ht (c, q, i, j) so that ht (c, q, i, j)=1if values of the quantization indexes Q(c, i, j) are q and otherwise,ht(c, q, i, j)=0, and prepares the histogram hc(q) based on suchdefinition.

$\begin{matrix}{{{hc}(q)} = {\sum\limits_{i}{\sum\limits_{j}{{ht}\left( {c,q,i,j} \right)}}}} & (1)\end{matrix}$

Next, the non-zero transform coefficient distribution estimating section524 in this example approximates the prepared histogram hc(q) by theLaplace distribution, taking this Laplace function as a distributionfunction of transform coefficients T.

An equation of the Laplace distribution can be expressed as follows:

$\begin{matrix}{{L(x)} = {\frac{1}{\sqrt{2}\sigma}{\exp\left( \frac{{- \sqrt{2}}{x}}{\sigma} \right)}}} & (2)\end{matrix}$

The non-zero transform coefficient distribution estimating section 524can obtain the distribution function of transform coefficients T bycalculating σ in Equation (2).

First, the non-zero transform coefficient distribution estimatingsection 524 normalizes the prepared histogram hc(q) with the width D(c)of a quantization interval and the total number of quantization indexes,and transforms the normalized histogram hc(q) into a probability densityfunction fhc(x). Specifically, the non-zero transform coefficientdistribution estimating section 524 transforms the histogram hc(q) intothe probability density function fhc(x) according to the followingequation.

$\begin{matrix}{{{fhc}(x)} = \frac{{hc}(q)}{{D(c)} \times {\sum\limits_{q}{{hc}(q)}}}} & (3)\end{matrix}$

Here, (q−0.5)×D(c)<x≦(q+0.5)×D(c)

Next, the non-zero transform coefficient distribution estimating section524 calculates the Laplace function approximating the histogram hc(q).

FIG. 6 is a diagram illustrating a histogram h and a distributionfunction L (Laplace function).

As shown in FIG. 6, the non-zero transform coefficient distributionestimating section 524 may find a to make a difference (an areadifference in this example) between the Laplace function L(x) and thehistogram fhc(x) as small as possible.

The following error function Err(σ) is defined as a function to estimate‘making the difference as small as possible’.

$\begin{matrix}{{{Err}(\sigma)} = {\sum\limits_{q}{{\int_{{({q - 0.5})} \times {D{(c)}}}^{{({q + 0.5})} \times {D{(c)}}}{\left\{ {{L(x)} - {{fhc}(x)}} \right\} {x}}}}}} & (4)\end{matrix}$

This error function Err(σ) is a function of summing absolute vales ofdifferences of areas of the probability density functions obtained forthe quantization index values q. As values of the error function Err(σ)become small, it can be said that the histogram fhc(x) approaches theLaplace function L(x). The non-zero transform coefficient distributionestimating section 524 obtains σ to minimize the error function Err(σ)through a numerical calculation.

The zero transform coefficient distribution estimating section 526estimates frequency distribution of zero transform coefficients based onthe frequency distribution of other transform coefficients estimated bythe non-zero transform coefficient distribution estimating section 524according to instructions from the zero determining section 522.

That is, the zero transform coefficient distribution estimating section526 can estimate the frequency distribution only if the histogram has ameaningful shape, but cannot estimate the shape of the frequencydistribution when the histogram in which all values of frequency arezero is prepared.

For that reason, the zero transform coefficient distribution estimatingsection 526 estimates the shape of the Laplace distribution in which allquantization indexes of the transform coefficient kind c are zero, usinganother obtained distribution data (σ in this example), according to amethod to be described below.

In this example, since the decoding process in the JPEG is described byway of example, transform coefficient kinds are arranged in atwo-dimensional 8×8 matrix.

Here, values of σ are arranged in a two-dimension, corresponding to(1, 1) to (8, 8) components of DCT coefficients, as shown in FIG. 7A.That is, values of σ corresponding to transform coefficients having (x,y) components are represented by σ(x, y).

For example, σ(1, 1) is a value of σ having a DC component and σ(8, 8)is a value of σ of a transform coefficient representing the highest ACcomponent. In this example, however, since the non-zero transformcoefficient distribution estimating section 524 and the zero transformcoefficient distribution estimating section 526 cannot approximate thevalue of σ corresponding to the DC component by the Laplacedistribution, this value of σ is not used for the estimation of valuesof σ.

In this example, it is assumed that σ(x, y) is a function on an x-yplane. The zero transform coefficient distribution estimating section526 determines this function σ(x, y) using the values of σ alreadyobtained (that is, the values of σ calculated by the non-zero transformcoefficient distribution estimating section 524) and estimates thevalues of σ corresponding to the zero transform coefficients.

Specifically, the zero transform coefficient distribution estimatingsection 526 approximates the function σ(x, y) by a two-dimensionalexponential function. That is, σ(x, y)=Cexp(−ax−by).

This corresponds to that the zero transform coefficient distributionestimating section 526 approximates the values of a by the exponentialfunction, as shown in FIG. 7(B).

The zero transform coefficient distribution estimating section 526calculates parameters C, a and b in the equation of σ(x, y)=Cexp(−ax−by)to determine the approximation function σ(x, y), and calculates thevalues of σ corresponding to the zero transform coefficients using thedetermined approximation function σ(x, y).

Here, σ(x, y) already obtained is set as σ(x(u), y(u)). Here, (x(u),y(u)), where u=1, 2, . . . , U, is a coordinate of the already obtainedσ.

In addition, since all quantization indexes are zero, values of σ whichcould not be obtained are set as σ(x(v), y(v)). Here, v=1, 2, . . . , V,and U+V=63.

First, the zero transform coefficient distribution estimating section526 determines C, a and b using σ(x(u), y(u)) (u=1, 2, . . . , U).

As a preparation for this, both sides of the equation of σ(x,y)=Cexp(−ax−by) are changed into a logarithmical form as follows:

log σ(x,y)=log C−ax−by

Next, σ(x(u), y(u)) is substituted into the logarithmical equation. Thatis,

log σ(x(u),y(u))=log C−ax(u)−by(u)

Here, since u=1, 2, . . . , U, the above equation is a calculation of amatrix as follows:

$\begin{matrix}{{\begin{pmatrix}{- {x(1)}} & {- {y(1)}} & 1 \\{- {x(2)}} & {- {y(2)}} & 1 \\\ldots & \ldots & \ldots \\{- {x(U)}} & {- {y(U)}} & 1\end{pmatrix}\begin{pmatrix}a \\b \\{\log \; C}\end{pmatrix}} = \begin{pmatrix}{\log \; {\sigma \left( {{x\; (1)},{y(1)}} \right)}} \\{\log \; {\sigma \left( {{x(2)},{y(2)}} \right)}} \\\ldots \\{\log \; {\sigma \left( {{x(U)},{y(U)}} \right)}}\end{pmatrix}} & (5)\end{matrix}$

Further, the above matrix can be solved as typical simultaneousequations if U=3. Also, the above matrix can be solved using the leastsquare method if U>3.

In this manner, the zero transform coefficient distribution estimatingsection 526 can obtain the parameters a, b and C by solving the matrix.

Next, the zero transform coefficient distribution estimating section 526calculates values of σ corresponding to the zero transform coefficientsby substituting x(v) and y(v) corresponding to the zero transformcoefficients into the equation of σ(x(v), y(v))=Cexp(−ax(v)−by(v)).

Furthermore, the zero transform coefficient distribution estimatingsection 526 may correct σ(x, y) to monotonously decrease with respect tox and y in order to obtain more pertinent estimation values of σ. Thatis, when σ(x(v), y(v)) is obtained with assuming that the equation ofσ(x(v), y(v))=Cexp(−ax(v)−by(v)), σ(x(v), y(v)) is made smaller than orequal to σ(x, y) whose coordinate (x, y) is less than that of σ(x(v),y(v)). Specifically, the zero transform coefficient distributionestimating section 526 makes an correction using the following equation.

σ(x(v),y(v))=min{σ(x(v)−1,y(v)),σ(x(v),y(v)−1),σ(x(v)−1,y(v)−1)}

[Details of Random-Number Generating Section]

Next, the random-number generating section 560 (FIG. 4) will bedescribed in more detail.

The random-number generating section 560 applies a variabletransformation, according to the quantization indexes Q(c, i, j) to beprocessed, to the distribution function fc(x) input from thedistribution estimating section 520 (that is, the function correspondingto the values of σ (distribution data) calculated by the non-zerotransform coefficient distribution estimating section 524 or the zerotransform coefficient distribution estimating section 526).

Specifically, assuming that the quantization indexes Q(c, i, j)=q and arange of transform coefficients T(c, i, j) in which the quantizationindexes Q(c, i, j)=q is d1 to d2, the random-number generating section560 generates the following function fcq(x).

$\begin{matrix}{{{fcq}(x)} = \left\{ \begin{matrix}\frac{{fc}\left( {{\frac{{d\; 2} - {d\; 1}}{{\alpha \; \max} - {\alpha \; \min}}\left( {x - {\alpha \; \min}} \right)} - {d\; 1}} \right)}{\int_{d\; 1}^{d\; 2}{{{fc}(t)}{t}}} & {{\alpha \; \min} \leq x < {\alpha max}} \\0 & {otherwise}\end{matrix} \right.} & (6)\end{matrix}$

This function fcq(x) is a probability density function corresponding tothe transform coefficient kind c and the transform coefficients of thequantization indexes q. In addition, this probability density functionis obtained by transforming the range of d1 to d2 into a range of αminto αmax. A method of determining αmin and αmax may be considered invarious ways as follows; for example, (1) a method of setting αmin andαmax as preset constant values when αmin=−αmax, (2) a method of using apreset constant value β and setting a value of (αmax−αmin) to be aquantization step size×β=αmax−αmin when αmin=−αmax, (3) a method ofsetting a upper limit value Dmax such that a value of the quantizationstep size×β is not larger than the preset constant value and determiningαmin and αmax with αmax−αmin=min{the quantization step size×β, Dmax}, inaddition to the above method (2), and (4) a method of setting a value of(αmin+αmax)/2 to be expected values of α (E(αTcq)) in modifications tobe described).

The random-number generating section 560 generates random numbers α,which match the probability density function fcq(x).

The random numbers α are used as correction factors cc for calculationof the dequantization values by the dequantization-value outputtingsection 590 (FIG. 4).

Further, although the random numbers α are first generated and then thedequantization values R are obtained in this example, R may be directlygenerated as the random numbers. That is, the random-number generatingsection 560 may generate the random numbers to match the followingprobability density function fcq(x) and the dequantization-valueoutputting section 590 may output the random numbers as thedequantization values to the inverse transforming section 60.

$\begin{matrix}{{{fcq}(x)} = \left\{ \begin{matrix}\frac{{fc}(x)}{\int_{d\; 1}^{d\; 2}{{fc}{t}}} & {{d\; 1} \leq x < {d\; 2}} \\0 & {otherwise}\end{matrix} \right.} & (7)\end{matrix}$

Next, a method of generating the random numbers to match the probabilitydensity function fcq(x) will be described. As such a random numbergenerating method, for example, an inverse function method, which isdisclosed in ‘Knowledge of random number’ (Wakimoto Kazumasa, MorikitaShuppan Co., Ltd., pp. 61-64), may be applied.

Now, an example for this will described.

First, the following function Fcq(x) is obtained.

$\begin{matrix}{{{Fcq}(x)} = {\int_{\alpha \; \min}^{x}{{{fcq}(t)}{t}}}} & (8)\end{matrix}$

Next, an inverse function F⁻¹cq(x) of Fcq(x) is obtained. Subsequently,random numbers X in an interval [0, 1] are generated from a uniformrandom generator.

Finally, when α=F⁻¹cq(x), the random numbers a to match fcq(x) can begenerated. Where, F⁻¹cq(x) is the inverse function of Fcq(x).

In addition, if fc(x) has the Laplace distribution, the functionF⁻¹cq(x) may be fixed in a simple form in advance. Here, for the sake ofsimplicity, fcq(x) is normalized in advance. Further, αmin=−αmax. Thatis, fcq(x)=Cexp(−s|x|) (where, αmin≦x≦αmax).

When q>0, since x≦0, fcq(x)=Cexp(−sx).

Accordingly,

Fcq(x)=−(C/s) {exp(−sx)−exp(−sαmin)}

F⁻¹cq(x)=−(1/s) log {exp (−sαmin)−sx/C}

When q<0, since x<0, similarly,

F⁻¹cq(x)=(1/s) log {exp(sαmin)+sx/C}

Moreover, when q=0 and x<0,

F⁻¹cq(x)=(1/s) log {exp(sαmin)+sx/C}

When q=0 and x≧0,

$\begin{matrix}\begin{matrix}{{{Fcq}(x)} = {{\int_{\alpha \; \min}^{0}{C\; {\exp ({sx})}{a}}} + {\int_{0}^{x}{C\; {\exp \left( {- {st}} \right)}{t}}}}} \\{= {0.5 - {\frac{C}{s}{\exp \left( {- {sx}} \right)}} + \frac{C}{s}}}\end{matrix} & (9)\end{matrix}$

Accordingly, F⁻¹cq(x)=−(1/s) log {1−s(x−0.5)/C}

In this manner, the random-number generating section 560 can generatesthe random numbers as the correction factors α appropriate for thedistribution of transform coefficients, based on the distribution datainput from the distribution estimating section 520.

[Correcting Section]

FIGS. 8A and 8B are schematic diagrams illustrating correction by thecorrecting section 580.

As shown in FIG. 8A, the correcting section 580 shifts (a3) thedistribution of dequantization values to match an estimated expectedvalue (a1) of the transform coefficients T to an expected value (a2) ofthe dequantization values.

Further, as shown in FIG. 8B, when the distribution of dequantizationvalues (the correction factors α in this example) deviates from thequantization interval d1 to d2 (in this example, a case (b1) where thedistribution of dequantization values deviates from the range of a (αminto αmax)), the correcting section 580 makes the distribution smalltoward the expected value of the dequantization values (the correctionfactors α) without moving the expected value (b2).

Furthermore, in this example, the correcting section 580 makes theabove-mentioned correction for the correction factors α input from thedequantization-value estimating section 500.

[Entire Operation]

Next, the entire operation of the decoding apparatus 2 (the decodingprogram 5) will be described.

FIG. 9 is a flowchart of a decoding process S10 executed by the decodingprogram 5 (FIG. 4). In this example, a case where the code data (of theJPEG) of image data are input will be described by way of examples.

As shown in FIG. 9, in Step S100, the entropy decoding section 40 (FIG.4) generates quantization indexes for each block (8×8 block) by decodingthe input code data and outputs the generated quantization indexes foreach block to the dequantizing section 50.

In Step S105, the distribution estimating section 520 estimates thedistribution of transform coefficients T for each transform coefficientkind, based on the plurality of quantization indexes input from theentropy decoding section 40.

Specifically, when quantization indexes corresponding to an image of asingle page are input from the entropy decoding section 40 to the zerodetermining section 522 (FIG. 5) provided in the distribution estimatingsection 520, the zero determining section 522 classifies the inputquantization indexes into transform coefficient kinds and determines asto whether the classified quantization indexes correspond to the zerotransform coefficients or the non-zero transform coefficients.

The non-zero transform coefficient distribution estimating section 524(FIG. 5) prepares the histogram hc(q) of the quantization indexes (thatis, the histogram for each transform coefficient kind c) for each groupof quantization indexes corresponding to the non-zero transformcoefficients and calculates the Laplace function L (that is, values ofσ) approximating the histogram hc(q).

Further, the zero transform coefficient distribution estimating section526 (FIG. 5) approximates the frequency distribution calculated by thenon-zero transform coefficient distribution estimating section 524 by anexponential function and estimates the frequency distribution of zerotransform coefficients (that is, values of σ) using this exponentialfunction.

In Step S110, the dequantizing section 50 (FIG. 4) sets the inputquantization indexes to a given quantization index in order.

The dequantization-value estimating section 500 (FIG. 4) extractsneighboring quantization indexes Q(c, i+m, j+n) (−1≦m≦1 and −1≦n≦1 inthis example) around the given quantization index Q(c, i, j). Theextracted neighboring quantization indexes are quantization indexes ofthe same transform coefficient kind c in 3×3 blocks around the givenblock and have a 3×3 matrix.

In Step S115, the dequantizing-value estimating section 500 prepares adifference matrix P by performing the following calculation using theextracted neighboring quantization indexes and the given quantizationindex.

P(m,n)=Q(c,i+m,j+n)−Q(c,i,j)

That is, the dequantizing-value estimating section 500 calculates adifference value between a value of the given quantization index andvalues of the neighboring quantization indexes.

Next, the dequantizing-value estimating section 500 compares an absolutevalue |P(m, n)| of each difference value included in the differencematrix P with a threshold TH (for example, 1), and sets a differencevalue P(m, n) larger than the threshold TH as 0 (threshold process).That is, if a difference between the neighboring quantization indexvalue and the given quantization index value is larger than thethreshold, the dequantizing-value estimating section 500 removes such aneighboring quantization index value as a non-correlation signal.

In Step S120, the dequantizing section 50 (FIG. 4) determines as towhether or not dequantization values can be estimated for the givenquantization index.

Specifically, if the given quantization index and all components of thedifference matrix P having been subjected to the threshold process are 0(for example, if values in all neighboring quantization indexes(quantization indexes of neighboring blocks) are equal, or if allneighboring quantization indexes are removed as non-correlation signals,etc), the dequantizing section 50 determines that the estimation ofdequantization values is impossible. Otherwise, the dequantizing section50 determines that the estimation of dequantization values is possible.

If the dequantizing section 50 determines that the estimation ofdequantization values (in this embodiment, estimation of the correctionfactors α) is possible, the process proceeds to Step S115. If thedequantizing section 50 determines that the estimation of dequantizationvalues is impossible, the process proceeds to Step S120.

In Step S125, the dequantization-value estimating section 500 calculatescorrection factors αycq, using a 3×3 filter kernel K(m, n) shown in FIG.10, by performing a convolution operation for the difference matrix Phaving been subjected to the threshold process. Accordingly, even whenvalues of the given quantization indexes are equal, if neighboringquantization indexes around the given quantization indexes aredifferent, the calculated correction factors αycq have different values.

In addition, a filter shown in FIG. 10 has a low pass characteristic.

In Step S130, the random-number generating section 560 generates randomnumbers according to the distribution data input from the distributionestimating section 520 for the given quantization index and outputs thegenerated random numbers to the dequantization-value outputting section590 as the correction factors α.

Specifically, the random-number generating section 560 selects adistribution corresponding to the given quantization index from amongthe distributions estimated by the non-zero transform coefficientdistribution estimating section 524 and the zero transform coefficientdistribution estimating section 526, generates random numbers to matchthe selected distribution, and outputs the generated random numbers tothe dequantization-value outputting section 590 as the correctionfactors α.

In Step S135, the dequantizing section 50 determines as to whether ornot the correction factors α are generated for all quantization indexes.If it is determined that the correction factors α are generated for allquantization indexes, the process proceeds to Step S140. Otherwise, theprocess returns to Step S110 where a next quantization index is taken asa given quantization index to be processed.

In Step S140, the expected-value estimating section 540 calculatesexpected values E(αTcq) of the probability density function for eachcombination of the transform coefficient kind and the quantizationindexes based on the distribution data input from the distributionestimating section 520, and outputs the calculated expected valuesE(αTcq) to the correcting section 580.

In Step S145, the correcting section 580 classifies the correctionfactors α calculated by the dequantization-value estimating section 500for each transform coefficient kind and each quantization index, andcalculates the minimum value, the maximum value and a mean value of theclassified correction factors α.

Next, the correcting section 580 compares the expected values E(αTcq)input from the expected-value estimating section 540 with the calculatedmean value for each combination of the transform coefficient kind andthe quantization indexes, and shifts a group of correction factors αycqclassified into combinations of the transform coefficient kind and thequantization indexes such that the expected values E(αTcq) become equalto the mean value (shift correction).

Further, the correcting section 580 determines as to whether or not thegroup of correction factors α having been subjected to the shiftcorrection falls within a range of −0.5 to 0.5. If it is determined thatthe group of correction factors α does not fall within the range, arange correction to make the range of the group of correction factorsαycq fall within the range of −0.5 to 0.5 is performed without changingthe mean value of the group of correction factors αycq.

In Step S150, the dequantization-value outputting section 590 (FIG. 4)calculates a dequantization value Ry to be applied, based on the givenquantization index Q and the correction factors α input from thecorrecting section 580 or the correction factors α input from therandom-number generating section 560, and outputs the calculateddequantization value Ry to the inverse transforming section 60.

Specifically, the dequantization-value outputting section 590 in thisexample calculates the dequantization value Ry by performing thefollowing calculation.

Ry(c,i,j)={Q(c,i,j)+α(c,i,j)}×D(c)

In Step S155, the inverse transforming section 60 (FIG. 4) performs aninverse transform (an inverse DCT in this example) using thedequantization value (approximate transform coefficient) input from thedequantizing section 50 to generate a decode image H.

As described above, the decoding apparatus 2 in this embodimentestimates the distribution of transform coefficients based on thequantization indexes, generates the random numbers to match theestimated distribution, and generates the dequantization values based onthe generated random numbers.

Accordingly, since the frequency distribution of dequantization valuesbecomes close to the frequency distribution of transform coefficients, adecoded image having higher reproducibility can be expected.

The configuration including the dequantization-value estimating section500 for calculating the correction factors α using the neighboringquantization index values and the random-number generating section 560for generating the random numbers, which matching the distribution ofquantization indexes, as the correction factors α has been described inthe above embodiment. However, the dequantization-value estimatingsection 500 is not essential. That is, the random-number generatingsection 560 may generate the correction factors α for all quantizationindexes.

[First Modification]

Next, a first modification will be described.

Although the distribution of transform coefficients is estimated by theLaplace distribution in the above embodiment, a polygonal-lineapproximation may performed for the distribution of transformcoefficients in the first modification, as shown in FIG. 11.

For example, assuming that αmid is defined as αmin+αmax, the non-zerotransform coefficient distribution estimating section 524 (FIG. 5) makesan estimation by approximating the probability density function by astraight line (polygonal line) connecting αmin, αmid, and αmax.

However, when the quantization index value q has an AC component andapproaches 0, the linear approximation is difficult to be achieved.Therefore, the non-zero transform coefficient distribution estimatingsection 524 employs another approximation method. More specifically, thenon-zero transform coefficient distribution estimating section 524alternates between a plurality of approximation methods based on athreshold TH1, which is a positive integer. That is, assuming that q isa quantization index value,

a first linear approximation is performed when |q|>TH1,

a second linear approximation is performed when |q|=TH1, and

a Laplace distribution approximation (described in the above embodiment)is performed when |q|<TH1.

Although this modification changes between the linear approximation andthe Laplace distribution approximation according to the quantizationindex values q, the first linear approximation may be applied for allvalues of q.

The first and second linear approximations are the polygonal-lineapproximations as shown in FIG. 11A.

To begin with, the first linear approximation will be described.

As shown in FIG. 11B, the non-zero transform coefficient distributionestimating section 524 considers a uniform function fk(α), whichsatisfies that fk(α)=hc(q) (where, αmin≦α≦αmax). This uniform functionis then approximated by a polygonal line.

The non-zero transform coefficient distribution estimating section 524estimates values of fk(αmin) and fk(αmax) using neighboring histogramshc(q−1) and hc(q+1) shown in FIG. 11A.

For example, a value of fk(αmax) is estimated as follows.

First, the non-zero transform coefficient distribution estimatingsection 524 determines a position of a point A as shown in FIG. 11C.Setting the value of fk(αmax) between hc(q) and hc(q+1) is reasonable.For example, it is preferable that fk(αmax)=(hc(q)+hc(q+1))/2.

In this example, a point internally dividing an interval between hc(q)and hc(q+1) by a ratio of hc(q):hc(q+1) is employed as the position ofpoint A.

This is preferable because a value of point A can become sufficientlysmall when a frequency value hc(q) of the given quantization index isless than a frequency value hc(q+1) of a neighboring quantization indexor the frequency value hc(q) of the given quantization index approacheszero.

At that time, the non-zero transform coefficient distribution estimatingsection 524 can calculates fk(αmax) according to the following equation.

fk(αmax)=2×hc(q)×hc(q+1)/(hc(q)+hc(q+1))

Similarly, the non-zero transform coefficient distribution estimatingsection 524 can calculates fk(αmin) according to the following equation.

fk(αmin)=2×hc(q)×hc(q−1)/(hc(q)+hc(q−1))

Next, the non-zero transform coefficient distribution estimating section524 estimates a value of fk(αmid). Here, shapes of the neighboringhistograms are classified into two kinds, which are shown in FIGS. 11Dand 11E, respectively.

As shown in FIG. 11D, when the histograms (frequency values) increase ordecrease monotonously with respect to the quantization index values q,the non-zero transform coefficient distribution estimating section 524sets fk(αmid)=hc(q).

Further, as shown in FIG. 11E, when histograms (frequency values) doesnot increase or decrease monotonously with respect to the quantizationindex values q, the non-zero transform coefficient distributionestimating section 524 calculates fk(αmid) satisfying a condition offk(αmid)>hc(q) when hc(q) has the maximum value (peak), and calculatesfk(αmid) satisfying a condition of fk(αmid)<hc(q) when hc(q) has theminimum value (valley).

More specifically, the non-zero transform coefficient distributionestimating section 524 adds a mean value of a difference betweenfk(αmin) and fk(αmax). That is, fk(αmid) is calculated according to thefollowing equation.

fk(αmid)=hc(q)+(hc(q)−fk(αmin)+hc(q)−fk(αmax))/2

The non-zero transform coefficient distribution estimating section 524may transform the above-obtained function fk(x) into the probabilitydensity function. That is, the probability density function fcq(x) is asfollows:

$\begin{matrix}{{{fcq}(x)} = \left\{ \begin{matrix}\frac{{fk}(x)}{\int_{\alpha \; \min}^{\alpha \; \max}{{{fc}(t)}{t}}} & {{\alpha \; \min} \leq x < {\alpha \; \max}} \\0 & {otherwise}\end{matrix} \right.} & (10)\end{matrix}$

Next, the second linear approximation will be described. The secondlinear approximation is a linear approximation applied when |q|=TH1.

When q=TH1, the left side (a case of q=TH1−1) is approximated by theLaplace distribution unlike the first linear approximation. Therefore,it is desirable that values of fk(αmin) are considered so as to satisfycontinuity of the distribution. Accordingly, the non-zero transformcoefficient distribution estimating section 524 calculates fk(αmin)according to the following equation.

$\begin{matrix}\begin{matrix}{{{fk}\left( {\alpha \; \min} \right)} = {L\left( {\left( {{\alpha \; \min} + q} \right) \times {D(c)}} \right)}} \\{= {\frac{1}{\sqrt{2}\sigma}{\exp\left( \frac{{- \sqrt{2}}{{\left( {{\alpha \; \min} + q} \right) \times {D(c)}}}}{\sigma} \right)}}}\end{matrix} & (11)\end{matrix}$

In addition, fk(αmax) and fk(αmid) by the second linear approximationare calculated in the same manner as the first linear approximation.Further, the non-zero transform coefficient distribution estimatingsection 524 calculates fcq(x) using three numerical values, that is,fk(αmin), fk(αmax) and fk(αmid) in the same manner as the first linearapproximation.

[Second Modification]

The random numbers are generated for all quantization index values q inthe above embodiment.

The second modification shows an example where the random numbers aregenerated for part of the quantization index values q.

For example, the dequantization-value estimating section 500 cannotestimate the dequantization values only when all differences between thegiven quantization index value and the neighboring quantization indexvalues are 0. Since a number of quantization index values aredistributed in 0 as shown in the histogram of FIG. 6, the possibilitythat all differences between the given quantization index value and theneighboring quantization index values are 0 as described above becomeshigh when the given quantization index value is 0.

On the contrary, when the given quantization index values is not 0,since the neighboring quantization index values are very likely to be 0,the possibility that all differences between the given quantizationindex value and the neighboring quantization index values are 0 becomeslow.

As described above, the decoding program 5 in this modification appliesthe random numbers generated by the random-number generating section 560as the correction factors α (or the dequantization values) when thegiven quantization index value q is 0, and applies the correctionfactors α (or the dequantization values) generated by thedequantization-value estimating section 500 when the given quantizationindex value q is not 0.

[Third Modification]

The random-number generating section 560 generates the random numbers,which match the function fcq(x), in the above embodiment. Random numbersdifferent from the function fcq(x) are generated in the thirdmodification.

The function fcq(x) is a function distributed between a range of αmin toαmax. Accordingly, if the quantization step size D(c) is large, adistortion caused when random numbers deviated from the expected valueof α are generated is likely to become large.

Accordingly, the third modification limits the range of random numbersas described below.

That is, the random-number generating section 560 generates randomnumbers, which match the following probability density function fcq1(x).

$\begin{matrix}{{{fcq}\; 1(x)} = \left\{ \begin{matrix}\frac{{fcq}(x)}{\int_{{E{({\alpha \; {Tcq}})}} - d}^{{E{({\alpha \; {Tcq}})}} + d}{{{fcq}(t)}{t}}} & {{{E\left( {\alpha \; {Tcq}} \right)} - d} \leq x \leq {{E\left( {\alpha \; {Tcq}} \right)} + d}} \\0 & {{x < {{E\left( {\alpha \; {Tcq}} \right)} - d}},{> {{E\left( {\alpha \; {Tcq}} \right)} + d}}}\end{matrix} \right.} & (12)\end{matrix}$

where E(αTcq) is an expected value of fcq(x). In this manner, theexpected value may be calculated. Alternatively, for the sake of simplecalculation, it may be preferable that E(αTcq)=0. In addition, a rangeof d is limited such that αmin≦E(αTcq)−d and E(αTcq)+d≦αmax.

The above equation is an example of taking a generation probability inthe neighborhood as 0 using only a center shape (−d to d) of fcq(x). Bydoing so, since values deviated from the expected value are not output,it is possible to limit a square error.

[Fourth Modification]

In a fourth modification, random numbers different from fcq(x) aregenerated.

There may be a case where load on the random number generating processby the inverse function method as shown in the above embodiment isgreat.

Accordingly, the random-number generating section 560 in the fourthmodification generates uniform random numbers. Further, in thismodification, for the sake of convenience of description, a variance ofthe Laplace distribution when the distribution of transform coefficientsare estimated by the Laplace distribution and the transform coefficientkind c is estimated by the Laplace distribution is assumed as σ(c).

That is, the random-number generating section 560 in this modificationgenerates the random numbers according to the following probabilitydensity function fcq2(x).

fcq2(x)=1/(2βσ), if E(αTcq)−βσ≦x≦E(αTcq)+βσ

fcq2(x)=0, otherwise

Where E(αTcq) is the expected value of fcq(x). This expected value maybe calculated. Alternatively, for the sake of simple calculation, it maybe preferable that E(αTcq)=0.

Considering that E(αTcq)=0, the probability density function fcq2(x) isa uniform distribution function in a range of [−βσ, βσ]. A value β isset such that a range of [E(αTcq)−βσ, E(αTcq)+βσ] does not exceed [αmin,αmax].

The value β is a parameter to control disorder of a decoded image.Increase of β leads to increase of disorder of an image. Decrease of βleads to decrease of disorder of an image, however, results in an imagehaving visible block distortion.

[Fifth Modification]

While the case in which the invention is applied to the JPEG has beendescribed in the above embodiment and the above modifications, theinvention is not limited thereto. In the fifth modification, an examplein which the invention is applied to the JPEG2000 will be described.Hereinafter, a difference between the application of the invention tothe JPEG2000 and the application of the invention to the JPEG will bedescribed.

In the application of the invention to the JPEG2000, the range of α isas follows:

−1≦α1, when Q(c, i, j)=0

0≦r+α≦1, when Q(c, i, j)>0

−1≦−r+≦α0, when Q(c, i, j)<0

Further, in the JPEG2000, each transform coefficient exists in afrequency domain decomposed as shown in FIG. 12. Here, α(x, y) isdefined as follows. Where, NL is the number of decomposition levels of awavelet transform.

σ(1, 3)=σ of a coefficient of NHL

σ(3, 1)=σ of a coefficient of NLH

σ(3, 3)=σ of a coefficient of NHH . . .

σ(2, 6)=σ of a coefficient of (N−1)HL

σ(6, 2)=σ of a coefficient of (N−1)LH

σ(6, 6)=σ of a coefficient of (N−1)HH . . .

That is, it may be generalized as follows.

σ(2^(N) ^(L) ^(−N),3×2^(N) ^(L) ^(−N))=σ of nHL

σ(3×2^(N) ^(L) ^(−N), 2^(N) ^(L) ^(−N))=σ of nLH

σ(3×2^(N) ^(L) ^(−N), 3×2^(N) ^(L) ^(−N))=σ of nHH  (13)

The above values σ may be standard deviations of simple signals or maybe results caused by the estimation of the Laplace distribution asdescribed in the above embodiment. As shown in FIG. 12, the values σ arerelatively arranged on an x-y plane around a range on a two-dimensionalfrequency domain of each transform coefficient.

The zero transform coefficient distribution estimating section 526calculates the values σ corresponding to the zero transform coefficientsby approximating the values σ by an exponential function.

Second Embodiment

Next, a second embodiment will be described.

In the second embodiment, a distribution determining method differentfrom that in the first embodiment will be described. More specifically,in the second embodiment, the values σ are obtained by calculating astandard deviation of the quantization indexes Q(c, i, j) for eachquantization coefficient kind c. Meanwhile, the decoding program 5 inthe second embodiment has a configuration shown in FIG. 4.

The distribution estimating section 520 in the second embodimentestimates a using an established function F₊(x, σ) or F⁻(x, σ). F₊(x, σ)or F⁻(x, σ), which is an integral function of the Laplace distribution,is expressed by the following equation.

$\begin{matrix}{{{F_{+}\left( {x,\sigma} \right)} = {\frac{1}{2}\left\lbrack {1 - {\exp\left( {{- \frac{\sqrt{2}}{\sigma}}x} \right)}} \right\rbrack}}{{F_{-}\left( {x,\sigma} \right)} = {\frac{1}{2}\left\lbrack {1 - {\exp\left( {\frac{\sqrt{2}}{\sigma}x} \right)}} \right\rbrack}}} & (14)\end{matrix}$

This function can express σ as a positive function of x and y for anequation having a form of y=F₊(x, σ) or F⁻(x, σ). That is, the standarddeviation σ can be obtained without performing a numerical calculation(repeated calculations). This gives an advantage of this embodiment overthe first embodiment.

Here, a normalized histogram of the quantization indexes q is set asH(q).

In this embodiment, given any integer N, the standard deviation σ, whichmakes the sum of normalized histograms having a range of thequantization indexes q from −N to N equal to a corresponding integralvalue of the Laplace distribution, is obtained.

When the quantization step size is D, a range of coefficients in which qfalls within the range of −N to N is −(2N+1)D/2 to (2N+1)D/2 in theJPEG.

Making the sum of normalized histograms having the range of thequantization indexes q from −N to N equal to the corresponding integralvalue of the Laplace distribution can be achieved according to thefollowing equation.

$\begin{matrix}{{{F_{+}\left( \frac{\left( {{2N} + 1} \right)D}{2} \right)} + {F_{-}\left( {- \frac{\left( {{2N} + 1} \right)D}{2}} \right)}} = {\sum\limits_{q = {- N}}^{N}{H(q)}}} & (15)\end{matrix}$

From the symmetrical property of the Laplace distribution, the aboveequation can be transformed into the following equation.

$\begin{matrix}{{2{F_{+}\left( \frac{\left( {{2N} + 1} \right)D}{2} \right)}} = {\sum\limits_{q = {- N}}^{N}{H(q)}}} & (16)\end{matrix}$

Solving Equation 16 for σ, the following equation is obtained.

$\begin{matrix}{\sigma = {- \frac{\left( {{2N} + 1} \right)D}{\sqrt{2}{\log\left\lbrack {1 - {\sum\limits_{q = {- N}}^{N}{H(q)}}} \right\rbrack}}}} & (17)\end{matrix}$

The distribution estimating section 520 obtains the standard deviation σusing Equation (17).

[First Modification]

Hereinafter, modifications of the second embodiment will be described.

As a first modification, an application of the second embodiment to theJPEG2000 will be described. In the JPEG2000, a range of coefficients inwhich q has a range of −N to N is −(N+1)D to (N+1)D. Making the sum ofnormalized histograms having the range of q from −N to N equal to thecorresponding integral value of the Laplace distribution can be achievedaccording to the following equation.

$\begin{matrix}\begin{matrix}{{{F_{+}\left( {\left( {N + 1} \right)D} \right)} + {F_{-}\left( {{- \left( {N + 1} \right)}D} \right)}} = {2{F_{+}\left( {\left( {N + 1} \right)D} \right)}}} \\{= {\sum\limits_{q = {- N}}^{N}{H(q)}}}\end{matrix} & (18)\end{matrix}$

[Second Modification]

While the integer N is a preset value (that is, an applied value) in thesecond embodiment, the decoding program 5 sets an appropriate integer Nin a second modification.

Here, a method of determining the integer N as a linear function of themaximum value of the quantization indexes q will be described by way ofexamples.

First, integers a and b are prepared in advance.

(1) The maximum value of absolute values of the quantization indexes qis assumed as qM. That is, it is assumed that qM=max {|qmax|, |qmin|}.

(2) When qM is 0, no process is performed.

(3) N is obtained with N=min{qM−1, round(a×qM+b)}

In these equations, max {A, B} represents a function to output largerone of A and B and min{A, B} represents a function to output smaller oneof A and B. In addition, the minimum value and maximum value of q areassumed as qmin and qmax, respectively. round( ) represents a roundingoff process such as rounding-off or rounding out six and larger anddisregarding the remaining.

In the item (3), the reason why the minimum value of qM−1 and round(a×qM+b), which is an output value of the linear function, is taken isthat a cannot be obtained because a denominator of Equation (17) tocalculate a becomes 0 when N=qM. That is, the maximum of N is processedto become qM−1.

FIG. 13 is a diagram illustrating the configuration of the distributionestimating section 520 in the second embodiment in more detail.

A shown in FIG. 13, the distribution estimating section 520 includes afrequency distribution measuring section 532, a histogram normalizingsection 534, an N value acquiring section 536 and a standard-deviationestimating section 538.

In the distribution estimating section 520, the frequency distributionmeasuring section 532 measures a frequency distribution h(q) based oninput quantization indexes Q(i) (where, i=1, 2, . . . ). The frequencydistribution h(q) represents the number of values of the quantizationindexes Q(i), which are q.

Further, the frequency distribution measuring section 532 acquires themaximum value qM of the absolute values of the quantization index valuesq.

The histogram normalizing section 534 normalizes the frequencydistribution h(q) measured by the frequency distribution measuringsection 532 and generates a normalized histogram H(q).

The N value acquiring section 536 determines a value of N based on qMacquired by the frequency distribution measuring section 532.Specifically, the N value acquiring section 536 performs theabove-mentioned processes (1) to (3).

The standard-deviation estimating section 538 calculates the standarddeviation σ based on the value of N acquired by the N value acquiringsection 536, the normalized histogram H(q) generated by the histogramnormalizing section 534, and the quantization step size D input fromoutside.

An experimental result when the value of N is set as described above isshown in FIG. 14.

Various images are transformed by DCT and the maximum value of thequantization indexes after quantization is obtained. Also, a standarddeviation of transform coefficients of a original image is obtained, σis calculated using various values of N, and then, a value of N mostappropriate to estimate the standard deviation of the transformcoefficients is obtained. FIG. 14 shows a relationship between themaximum value of quantization values and the optimal value of N. Asshown in FIG. 14, there is a linear relationship between qM and theoptimal value of N.

Accordingly, a significant effect of obtaining N using the linearfunction of qM as in this method can be achieved.

Further, the value of N may be obtained with assuming that b=0 andN=min{qM−1, round(a×qM)}.

That is, as shown in FIG. 14, since the relationship between qM and theoptimal value of N is linear nearly passing an original point, b may belimited to zero.

In the above description, with the maximum value of the absolute valuesof the quantization indexes q as qM, N is obtained using the linearfunction of qM. However, it is not indispensable to set qM as themaximum value of the absolute values of the quantization indexes. qmaxor qmin may be also used as qM.

This is because the distribution of quantization indexes isapproximately symmetrical and a relationship of qmax=−qmin is almostestablished. That is, it is not necessary to use the maximum value ofthe absolute values.

Further, in the above description, the value of N is obtained byapplying the rounding off process using the round ( ) function to anoutput of the linear function. A rounding-down process or a rounding-outprocess may be employed in order to obtain the integer value N.

Furthermore, although a range of addition of the frequency distributionis −N to N in the above description, such a symmetrical range may not beemployed. For example, the range may be Nmin to Nmax, where Nmin≦0 andNmax≦0. For example, Nmin may be obtained as a function of qmin and Nmaxmay be obtained as a function of qmax.

[Third Modification]

In a third modification, estimation is made using a value of N, whichmakes an accumulated value of H(q) to be a certain value P(0<P<1). Thatis, the N value acquiring section 536 in the third modification obtainsN by giving a preset value P as expressed by the following equation.

$\begin{matrix}{N = {\underset{N}{argmin}{{\left\{ {\sum\limits_{q = {- N}}^{N}{H(q)}} \right\} - P}}}} & (19)\end{matrix}$

More specifically, the following operation is performed with using theconfiguration as shown in FIG. 13.

Data input to the distribution estimating section 520 is thequantization index Q(i) (where, I=1, 2, . . . ) and the quantizationstep size D.

The frequency distribution measuring section 532 measures the frequencydistribution h(q) based on the input quantization index Q(i). h(q)represents the number of values of quantization indexes Q(i), which areq.

At the same time, the frequency distribution measuring section 532acquires the maximum value qM of the absolute values of the quantizationindex values q.

Next, the histogram normalizing section 534 generates the normalizedhistogram H(q) based on the frequency distribution h(q) measured by thefrequency distribution measuring section 532.

The N value acquiring section 536 determines the value of N based on thenormalized histogram H(q) generated by the histogram normalizing section534 and qM acquired by the frequency distribution measuring section 532.In addition, the value P is a preset value.

More specifically, the N value acquiring section 536 determines thevalue of N according to the flow chart as shown in FIG. 15. Further, qmin the flow chart of FIG. 15 indicates the maximum value (qM) of theabsolute values of q. Furthermore, SUM indicates an accumulated value(accumulated frequency) of the frequency value H(q).

First, the N value acquiring section 536 sets SUM=H(0) and i=0 (S200).

Next, the N value acquiring section 536 sets N=1 and terminates theprocess (S210) if SUM≧P or qm=1 (S205: Yes) and otherwise (S205: No),increments a value of i by one and sets NewSUM=SUM+H(i)+H(−i) (S215).That is, the N value acquiring section 536 expands the range ofaccumulation of the frequency value by one in the left and right,respectively.

The N value acquiring section 536 sets N=i and terminates the process(S230) if NewSUM≧P (S220: Yes) and |NewSUM−P|≦|SUM−P| (S225: Yes). Also,the N value acquiring section 536 sets N=i−1 and terminates the process(S235) if NewSUM≧P (S220: Yes) and |NewSUM−P|>|SUM−P| (S225: No).

On the other hand, the N value acquiring section 536 sets N=qm−1 andterminates the process (S245) if NewSUM<P (S220: No) and i=qm−1 (S240:Yes). Also, the N value acquiring section 536 substitutes a value ofNewSUM into SUM (S250) and returns to S215 if NewSUM<P (S220: No) andunless i=qm−1 (S240: No).

The N value acquiring section 536 determines the value of N according tothe above process.

In addition, as shown in the above-described flow chart, since a valueof ΣH(q) is completed to be calculated, the N value acquiring section536 outputs the following equation, which is a result of addition of thenormalized histogram, to the standard-deviation estimating section 538.

$\begin{matrix}{\sum\limits_{q = {- N}}^{N}{H(q)}} & (20)\end{matrix}$

Using the equations shown in the second embodiment, thestandard-deviation estimating section 538 calculates the standarddeviation σ based on the value of N input from the N value acquiringsection 536, the result of addition of the normalized histogram H(q),and the quantization step size D input from outside.

As described above, an example of a result when the standard deviationis estimated is shown as below.

A root mean square error (RMSE) between the standard deviation oforiginal transform coefficients measured for various images, colorcomponents and quantization step sizes and the estimated standarddeviation is calculated.

When the value of N is determined according to the method shown in thesecond modification, RMSE is 4.176.

In addition, when the value of N is determined according to the methodshown in the third modification, RMSE is 4.033.

In this manner, the method shown in the above modifications can estimatethe standard deviation accurately.

Further, the method in this embodiment does not require a numericalcalculation as in the conventional technique, and accordingly, canstably perform a high-speed calculation without falling in a localsolution.

FIG. 16 shows the ratio of the standard deviation of original transformcoefficients measured for various images, color components andquantization step sizes to the estimated standard deviation. Since avertical axis represents a ratio of an estimated standard deviation/atrue standard deviation, a ratio approaching 1 indicates a goodperformance.

When a value of X is large, it can be seen from FIG. 16 that the secondmodification provides an excellent method showing good performance whilea conventional example shows poor performance with significantlydeteriorated values.

In addition, when the value of X is small, a method of the conventionalexample is relatively good. Moreover, when the value of X is large, thesecond or third modification is good.

[Fourth Modification]

Accordingly, in a fourth modification, the method disclosed in JP2004-80741 A (that is, the conventional example) is applied when thevalue of X=D/σ is small, and the second or third modification is appliedwhen the value of X is large.

In this case, since a true value of σ is unknown, the distributionestimating section 520 first estimates a with the same method as thesecond or third modification, evaluates a value of X using the estimatedσ, employs a estimated according to the method of the conventionalexample when X is smaller than a preset threshold, and employs σestimated according to the second or third modification when X is largerthan the preset threshold.

Further, the distribution estimating section 520 estimates a using thesecond or third modification when the maximum value of the quantizationindexes (or the maximum value of the absolute values) is smaller than apreset threshold and estimates a using the method of the conventionalexample when the maximum value of the quantization indexes (or themaximum value of the absolute values) is larger than the presetthreshold.

Furthermore, the distribution estimating section 520 may calculate andapply an intermediate value of σ between the value of σ calculated bythe method of the conventional example and the value of σ calculated bythe second or third modification.

That is, as shown in FIG. 16, since a value smaller than an actualstandard deviation is often calculated in the method of the conventionalexample and a value larger than the actual standard deviation is oftencalculated in the second or third modification, the distributionestimating section 520 can integrate these calculation results to takethe intermediate value of σ, thereby obtaining a value of σ closer tothe actual standard deviation.

More specifically, assuming that the standard deviation calculated bythe method of the conventional example is A and the standard deviationcalculated by the second or third modification is B, the distributionestimating section 520 calculates a final standard deviation σ accordingto the following equation. In this equation, a constant c is a presetvalue.

$\begin{matrix}{\sigma = \left( {AB}^{c} \right)^{\frac{1}{1 + c}}} & (21)\end{matrix}$

For example, when c=1, the final standard deviation σ becomes ageometrical mean of the standard deviation A and the standard deviationB. A result when c=1 is shown in FIG. 17. An estimated standarddeviation relative value on a vertical axis indicates a value of anestimated standard deviation value/a true standard deviation value.

As shown in FIG. 17, precision of estimation has been further improvedin the fourth modification.

For the fourth modification, RMSE=SQRT (a mean value of square ofdifference between the true standard deviation and the estimatedstandard deviation) is calculated. Where, SQRT( ) is a function tocalculate a square root.

RMSE in the conventional example: 2.26

RMSE in the fourth modification: 1.22

It can be seen from the above results that an efficiency (precision) ofthe fourth modification is excellent.

1. A distribution determining method comprising: adding values offrequency of the respective quantization indexes; and calculating atleast one of a variance of a Laplace distribution and a standarddeviation of the Laplace distribution so that a resultant value of theaddition of values of frequency is equal to an integral value when aLaplace distribution function is integrated so that an integral range ofa right side is equal to an integral range of a left side with using amaximum frequency position of the Laplace distribution as a reference.2. The distribution determining method according to claim 1, wherein thevalues of frequency of the respective quantization indexes are added sothat an addition range on a right side is equal to an addition range ona left side with using a maximum value of frequency in frequencydistribution of the quantization indexes as a reference.
 3. Thedistribution determining method according to claim 1, furthercomprising: determining a range within which the values of frequency areadded, on a basis of a maximum value of absolute values of thequantization indexes.
 4. The distribution determining method accordingto claim 3, further comprising: substituting the maximum value ofabsolute values of the quantization indexes into a preset linearfunction; rounding off a value obtained from the linear function toobtain an integer; and setting the obtained integer as the range withinwhich the values of frequency are added.